Helmholtz decomposition with a scalar Poisson equation in elastic anisotropic media

P-and S-wave separation plays an important role in elastic reverse-time migration.It can reduce the artifacts caused by crosstalk between different modes and improve image quality.In addition,P-and Swave separation can also be used to better understand and distinguish wave types in complex media.At present,the methods for separating wave modes in anisotropic media mainly include spatial nonstationary filtering,low-rank approximation,and vector Poisson equation.Most of these methods require multiple Fourier transforms or the calculation of large matrices,which require high computational costs for problems with large scale.In this paper,an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain.For 2D problems,the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation.Therefore,compared with existing methods based on pseudoHelmholtz decomposition operators,this method can significantly reduce the computational cost.Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly.

ON FREE WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA

The method of reverberation-ray matrix （MRRM） is extended and modified for the analysis of free wave propagation in anisotropic layered elastic media. A general, numerically stable formulation is established within the state space framework. The compatibility of physical variables in local dual coordinates gives the phase relation, from which exponentially growing functions are excluded. The interface and boundary conditions lead to the scattering relation, which avoids matrix inversion operation. Numerical examples are given to show the high accuracy of the present MRRM.

ELASTIC WAVEFIELD CALCULATION FOR HETEROGENEOUS ANISOTROPIC POROUS MEDIA USING THE 3-D IRREGULAR-GRID FINITE-DIFFERENCE

Basedonthe first-order Biot-equation with simplified coefficients,astaggered irregu- lar-grid finite difference method(FDM)is developed to simulate elastic wave propagation in 3-D heterogeneous anisotropic porous media.The ‘slow’P wave in porous media wave simulation is highly dispersive.Finer grids are needed to get a precise wavefield calculation for models with curved interface and complex geometric structure.Fine grids in a global model greatly increase computation costs of regular grids scheme.Irregular fine or coarse grids in local fields not only cost less computing time than the conventional velocity-stress FDM,but also give a more accu- rate wavefield description.A dispersion analysis of the irregular-grid finite difference operator has confirmed the stability and high efficiency.The absorbing boundary condition is used to elimi- nate artificial reflections.Numerical examples show that this new irregular-grid finite difference method is of higher performance than conventional methods using regular rectangular grids in simulating elastic wave propagation in heterogeneous anisotropic porous media.

Reflection and transmission of Laguerre Gaussian beam from uniaxial anisotropic multilayered media

Based on angular spectrum expansion and 4 × 4 matrix theory, the reflection and transmission characteristics of a Laguerre Gaussian （LG） beam from uniaxial anisotropic multilayered media are studied. The reflected and transmitted beam fields of an LG beam are derived. In the case where the principal coordinates of the uniaxial anisotropic media coincide with the global coordinates, the reflected and transmitted beam intensities from a uniaxial anisotropic slab and three-layered media are numerically simulated. It is shown that the reflected intensity components of the incident beam, especially the TM polarized incident beam, are smaller than the transmitted intensity components. The distortion of the reflected intensity component is more evident than that of the transmitted intensity component. The distortion of intensity distribution is greatly affected by the dielectric tensor and the thickness of anisotropic media. We finally extend the application of the method to general anisotropic multilayered media.

Propagation of Elastic Plane Waves in Homogeneous Anisotropic Media

Combining the linear transformation and the solution technique for the cubic equation, a general closed-form analytic solution for bulk waves in orthotropic anisotropic materials is obtained. This method is straightforward and general. Degenerated cases include transversely isotropic, cubic, and isotropic materials. Numerical computations are carried out on a fiber-reinforced composite plate modeled as a transversely isotropic media. The fibers are parallel to the top and bottom surfaces of the plate, and they are rotated counterclockwise around the plate normal through different angles. The two-dimensional slowness curves corresponding to different rotations are presented graphically. The wave propagation characteristics displayed in slowness surfaces for different fiber orientation are analyzed. Key words composite material - anisotropic media - wave propagation - slowness PASC 2001 0343.8 - 042 Project supported by the Science Foundation of Shanghai Municipal Commission of Education (Grant No. 03AK48)

Ray tracing of turning wave in elliptically anisotropic media with an irregular surface

Seismic ray tracing in anisotropic media with irregular surface is crucial for the exploration of the fine crustal structure. Elliptically anisotropic medium is the type of anisotropic media with only four independent elastic parameters. Usually, this medium can be described by only the vertical phase velocity and the horizontal phase velocity for seismic wave propagation. Model parameteri- zation in this study is described by flexible triangular grids, which is beneficial for the description of irregular surface with high degree of approximation. Both the vertical and horizontal phase velocities are defined in the triangular grids, respectively, which are used for the description of phase velocity distribution everywhere in the model by linear interpolation. We develop a shooting ray tracing method of turning wave in the elliptically anisotropic media with irregular surface. Runge-Kutta method is applied to solve the partial differential equation of seismic ray in elliptically anisotropic media. Linearly modified method is used for adjusting emergent phase angles in the shooting scheme. Numerical tests demonstrate that ray paths coincide well with analytical trajectories in trans- versely homogeneous elliptically anisotropic media. Seis- mic ray tracing results in transversely inhomogeneous elliptically anisotropic media demonstrate that our method is effective for further first-arrival tomography in ellipti- cally anisotropic media with an irregular surface.

Hydrothermal fluid circulation in anisotropic permeable media associated with discrete fractures

A proper form of the Rayleigh number, containing the geometric mean of the vertical and horizontal permeabilities was obtained. The critical value for the onset of stable convection was found. The results proved analytically and numerically that anisotropy in permeability resists the initiation of hydrothermal convection. The equivalence between homogeneously anisotropic media and multiply fractured media was also investigated. It was confirmed that multiply fractured models are comparable to anisotropic models as long as they have the same averaged horizontal or vertical permeabilities and other physical parameters.

Diffraction of Elastic Waves in Anisotropic Media and Dynamic Stress Concentration

The problems of diffraction of elastic waves in anisotropic media are investigated. The mathematical structures of the wave fields are found out by using the complex function method, and based on which the method for solving the problems of the first and second boundary value and the expressions of the dynamic stress concentraction factor are given.

A staggered-grid high-order finite-difference modeling for elastic wave field in arbitrary tilt anisotropic media

The paper presents a staggered-grid any even-order accurate finite-difference scheme for two-dimensional (2D), three-component (3C), first-order stress-velocity elastic wave equation and its stability condition in the arbitrary tilt anisotropic media; and derives a perfectly matched absorbing layer (PML) boundary condition and its stag- gered-grid any even-order accurate difference scheme in the 2D arbitrary tilt anisotropic media. The results of nu- merical modeling indicate that the modeling precision is high, the calculation efficiency is satisfactory and the absorbing boundary condition is better. The wave-front shapes of elastic waves are complex in the anisotropic media, and the velocity of qP wave is not always faster than that of qS wave. The wave-front triplication of qS wave and its events in both reflected domain and propagated domain, which are not commonly hyperbola, is a common phenomenon. When the symmetry axis is tilted in the TI media, the phenomenon of S-wave splitting is clearly observed in the snaps of three components and synthetic seismograms, and the events of all kinds of waves are asymmetric.

A NEW FORMULA FOR INDUCTANCE OF ANISOTROPIC MAGNETIC MEDIA

By solving the basic equations of magnetic field in the anisotropic magnetic mediain which the tensor μik is symmetric,an integral formula for anisotropic vector potential A isobtained.By which the characteristic formulae for self and mutual inductances are derived inscalar and tensor forms,and their transformation formula are also derived.Finally,the formulais checked by practical examples.

An eigen theory of static electromagnetic field for anisotropic media

Static electromagnetic fields are studied based on standard spaces of the physical presentation, and the modal equations of static electromagnetic fields for anisotropic media are derived. By introducing a new set of first-order potential functions, several novel theoretical results are obtained. It is found that, for isotropic media, electric or magnetic potentials are scalar; while for anisotropic media, they are vectors. Magnitude and direction of the vector potentials are related to the anisotropic subspaces. Based on these results, we discuss the laws of static electromagnetic fields for anisotropic media.

Finite element equations and numerical simulation of elastic wave propagation in two-phase anisotropic media

Based on Biot theory of two-phase anisotropic media and Hamilton theory about dynamic problem,finite element equations of elastic wave propagation in two-phase anisotropic media are derived in this paper.Numerical solution of finite element equations is given.Finally,Properties of elastic wave propagation are observed and analyzed through FEM modeling.

SCATTERING OF PLANE SH-WAVES ON SEMI-CANYON TOPOGRAPHY OF ARBITRARY SHAPE WITH LINGING IN ANISOTROPIC MEDIA

The purpose of this paper is to use the conforma mapping method[1]to analyzeand evaluate the ground displacement and scattering of incident SH-waves, on thesurface of semi-canyon topography of arbitrary shape with lining in anisotropic media.The problem to be solved can be reduced to the solution of an infinite algebraicequation set by using the method of full-space expansion of Fourier progression Usingthe mapping function and scattering theory to solve problems due to semi-canyon topography with lining is just like mapping the semi-cylindrical canyon of arbitraryshape into a cylindrical canyon in full-space.Moreover,it is far practical inengineering practice.From the computational examples,it is obvious that the variation of displacement amplitudes on the surface near the canyon topography is rather sharp. especially when the freqencies of incident SH-waves increase.

THE INTERACTION OF PLANE SH－WAVES AND NON－CIRCULAR CAVITY SURFACED WITH LINING IN ANISOTROPIC MEDIA

This is an expand of the complex function method in solving the problem of interaction of plane.SH-waves and non-circular cavity surfaced with linig in anisotropic media.the use the method similar to that incorporated in [2] added with Savin's method for solving stress concentration of non-circular cavity surfaced with lining in elasticity.Anisotropic media can be used ic simulate the conditions of thegeology.The solving proceeding for this problem can be processed conveniently in the manner similar to that introduced in [2].In this paper.as illustrated in example numerical studies have been done for a square cavity surfaced with lining in anisotropic media.

The Quasi-Static Approximation of Heat Waves in Anisotropic Thermo-Elastic Media

The equilibrium equations of anisotropic media, coupled to the heat conduction equations, are studied here based on the standard spaces of the physical presentation, in which an new thermo-elastic model based on the second law of thermodynamics is induced. The uncoupled heat wave equation for anisotropic media is deduced. The results show that the equation of heat wave is of the properties of dissipative waves. In final part of this paper, we discuss the propagation behaviour of heat waves for transversely isotropic media.

Sound-transparent anisotropic media for backscattering-immune wave manipulation

The scattering behavior of an anisotropic acoustic medium is analyzed to reveal the possibility of routing acoustic signals through the anisotropic layers with no backscattering loss. The sound-transparent effect of such a medium is achieved by independently modulating the anisotropic effective acoustic parameters in a specific order, and is experimentally observed in a bending waveguide by arranging the subwavelength structures in the bending part according to transformation acoustics. With the properly designed filling structures, the original distorted acoustic field in the bending waveguide is restored as if the wave travels along a straight path. The transmitted acoustic signal is maintained nearly the same as the incident modulated Gaussian pulse. The proposed schemes and the supporting results could be instructive for further acoustic manipulations such as wave steering, cloaking and beam splitting.

Study and application of PS-wave pre-stack migration in HTI media and an anisotropic correction method

Anisotropy correction is necessary during the processing of converted PS- wave seismic data to achieve accurate structural imaging, reservoir prediction, and fracture detection. To effectively eliminate the adverse effects of S-wave splitting and to improve PS- wave imaging quality, we tested methods for pre-stack migration imaging and anisotropic correction of PS-wave data. We based this on the propagation rules of seismic waves in a horizontal transverse isotropy medium, which is a fractured medium model that reflects likely subsurface conditions in the field. We used the radial （R） and transverse （T） components of PS-wave data to separate the fast and slow S-wave components, after which their propagation moveout was effectively extracted. Meanwhile, corrections for the energies and propagation moveouts of the R and T components were implemented using mathematical rotation. The PS-wave imaging quality was distinctly improved, and we demonstrated the reliability of our methods through numerical simulations. Applying our methods to three-dimensional and three-component seismic field data from the Xinchang-Hexingchang region of the Western Sichuan Depression in China, we obtained high-quality seismic imaging with continuous reflection wave groups, distinct structural features, and specific stratigraphic contact relationships. This study provides an effective and reliable approach for data processing that will improve the exploration of complex, hidden lithologic gas reservoirs.

Two Concepts in Optics of Anisotropic Dispersive Media and Polariton Case in Coordinate-Invariant Way

Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic media with material tensors for electric and magnetic induction which only depend on the frequency. The general algebraic form of the polarization vectors for the electric field and their one-dimensional projection operators is discussed without the degenerate cases of optic axis for which they become two-dimensional projection operators. Group velocity and diffraction coefficients in an approximate equation for the slowly varying amplitudes of beam solutions are calculated. As special case a polariton permittivity for isotropic media with frequency dispersion but without losses is discussed for the usual passive case and for the active case (occupation inversion of two energy levels that goes in direction of laser theory) and the group velocity is calculated. For this active case, regions of frequency and wave vector with group velocities greater than that of light in vacuum were found. This is not fully understood and due to large diffraction is likely only to realize in guided resonator form. The notion of “negative refraction” is shortly discussed but we did not find agreement with its assessment in the original paper.

Enhancing undulation of soft robots in granular media:A numerical and experimental study on the effect of anisotropic scales

Generating efficient locomotion in granular media is important,although it is difficult for robots.Inspired by the fact that sand vipers usually have saw-like scales,in this study,we design a soft undulation robot with tangential anisotropic friction to enhance the undulation performance of soft robots in granular media.A mathematical model was derived and numerical simulations were conducted accordingly to investigate the effectiveness of tangential friction anisotropy for undulation gait generation in granular media.In particular,we introduce a pseudo-rigid-body dynamics model consisting of links and joints while simulating the pneumatic actuation method to more closely approximate the response of soft robots.Moreover,a soft snake-like robot was fabricated,and its forward and reverse undulations were compared in two sets of controlled experiments.The consistency between the experimental results and the numerical simulations confirms that tangential anisotropic friction induces a propulsive effect in undulation,thereby increasing the robot's locomotion speed.This discovery provides new insights into the design of undulation robots in granular environments.2024 The Author(s).Published by Elsevier B.V.on behalf of Shandong University.This is an open access articleunder the CCBY license(http://creativecommons.org/licenses/by/4.0/).

Scattering Analysis of Bi-anisotropic Materials Using Fast Dipole Method

Electromagnetic scattering from inhomogeneous three-dimensional( 3D) bi-anisotropic scatterers is formulated in terms of the volume integral equation( VIE) method. Based on the volume equivalence principle,the VIE is represented in terms of a pair of coupled bi-anisotropic polarized volume electric and magnetic flux densities. The VIE is solved using the method of moments( MoM) combined with tetrahedral mesh. Then the fast dipole method( FDM) based on the equivalent dipole method( EDM) is extended to analyze the scattering of bi-anisotropic media by solving the VIE. Finally,some numerical results are given to demonstrate the accuracy of the developed method for the scattering analysis of the bi-anisotropic media.